Background

This document has nls (non-linear least squares) regression fits using the log-normal functional form to USFS FIA (United States Forest Service Forest Inventory & Analysis) biomass growth vs. stand age relationships. This functional form is commonly used in growth analyses, and permits a flexible shape to fit to data with an intermediate maximum (i.e., “hump” shaped) curve. As in our models of biomass growth vs. biomass, we use the mass balance biomass growth method for the plot biomass growth (\(G\)) calculation (briefly, plot biomass growth is a function of the change in plot biomass plus any losses due to mortality or harvest over time: \(G_{MB} = (\Delta B + M_t + C_t) / REMPER\), where \(\Delta B\) is change in plot biomass over a census interval ( \(\Delta B = B_{t + \Delta g} - B_t\) ), and \(M_t\) and \(C_t\) is the biomass of trees that died or were harvested, respectively, between two plot measurements. note: \(REMPER\) is time between two plot measurement intervals (FIA re-measurement period). For additional details see supplementary methods. Models are fitted separately by US ecoprovince.

Hypothetically, the entire functional form of the following non-linear model is considered: \(G = (1 + (yr-1990) \cdot tau/100) \times (1 - \alpha \cdot B_l) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\), where \(G\) is the plot level biomass growth calculated as the sum of tree biomass growth increments, \(B_l\) is the calculated proportion of biomass loss over the census interval, \(StdAge_{t1}\) is the FIA-estimated stand age at the first of two FIA plot tree censuses, and \(yr\) is the measurement year (all FIA data). Free parameters are \(\alpha\): the growth compensation of lost plot biomass, \(\tau\): the productivity trend, \(a\): the y-intercept of the curve, \(a +b\): the peak value of \(G\), \(c\): the \(StdAge_{t1}\) value at peak \(G\), and \(d\): the curve shape parameter.

Data have increasing variance in \(G\) with increasing \(StdAge_{t1}\), Thus, weighted nls is the best approach. We explore a few weighting options and found that proportional weighting can be achieved by weighting observations by \(\frac {1} {StdAge_{t1}^2}\) in equal-sample sized plot biomass bins (n=20 where applicable, else n=10) for each ecoprovince. These bins are also used to visualize data means in relation to nls model fit.

Model selection is done to determine the best fitting models, considering the inclusion of \(\alpha\): the biomass compensation effect due to lost biomass (natural mortality or harvest). Thus, the following two models are considered:

model 1: simple (tau) model \(G = (1 + (yr-1990) \cdot tau/100) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\)

model 3: model \(G = (1 + (yr-1990) \cdot tau/100) \times (1 - \alpha \cdot B_l) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\)

NOTE:

This document contains all \(G\) observations that meet our plot based filtering criteria:

  1. exclude FIA plots in plantation forests
  2. exclude FIA plots with multiple plot conditions (COND_PROG_UNADJ > 0.95)
  3. exclude FIA plots non-productive stands (i.e., those with less than 20 ft^3/acre/year timber producing capability; SITECLCD of 7)
  4. exclude FIA plots in non-stocked stands (i.e., those with STDSZCD of 5)
  5. exclude FIA plots in non-accessible areas (i.e., private lands etc., COND_STATUS_CD not equal to 1)
  6. exclude FIA plot visits that are not part of the annual inventories (which also includes FIA plot visits for Phase 3 ozone measurements)

Additionally, in an effort to clean up the data set, we have removed outlier observations, using a quantile threshold approach. We also calculated plot \(G\) using as biomass balance method (see supplementary methods), and the difference between the two methods. Accordingly, we define \(diff_G\) as the difference between tree incremental \(G\) and biomass balance \(G\). We excluded observations which meet the following criteria using a 0.5% quantile (\(QT\)):

  • case A: where the \(QT\) difference in tree incremental \(G\) is > biomass balance plot G (i.e., > 99.5% \(diff_G\) positive outliers)

  • case B: where the \(QT\) difference in tree incremental \(G\) is < mass balance plot G (i.e., < 0.5% \(diff_G\) negative outliers)

  • case C: where the \(QT\) difference in tree incremental \(G\) is > 0 (i.e., > 99.5% positive outliers)

  • case D: where the \(QT\) difference in tree incremental \(G\) is > 0 (i.e., < 0.5% negative outliers)

These data set cleaning criteria resulted in the exclusion of 1760 observations.

Below the model fitting procedure is implemented by ecoprovince:

211 - Northeastern Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   6869     2901.4                                
## 2   6817     2055.3 52 846.18  53.974 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 29461.42
## 2     2 26946.57
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.32590    0.17948   1.816   0.0694 .  
## alpha  0.64710    0.03386  19.114   <2e-16 ***
## a      0.00000    2.44054   0.000   1.0000    
## b      3.45262    2.42976   1.421   0.1554    
## c     31.41828    2.14731  14.631   <2e-16 ***
## d      2.77428    1.17757   2.356   0.0185 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5491 on 6817 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (54 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.

plotting 2

212 - Laurentian Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1  19351     8864.8                                 
## 2  18862     4850.3 489 4014.5  31.926 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 83315.42
## 2     2 70488.69
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.75653    0.19895   8.829  < 2e-16 ***
## alpha  0.76060    0.02177  34.931  < 2e-16 ***
## a      0.86040    0.14171   6.072 1.29e-09 ***
## b      1.40974    0.13886  10.152  < 2e-16 ***
## c     21.80997    0.54644  39.913  < 2e-16 ***
## d      1.96787    0.15867  12.402  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5071 on 18862 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (3847 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 45 rows containing missing values (`geom_point()`).

plotting 2

221 - Eastern Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   7319     3437.6                                
## 2   7255     2945.2 64  492.4  18.952 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 34070.15
## 2     2 32734.85
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.59046    0.14350  -4.115 3.92e-05 ***
## alpha  0.76479    0.04016  19.046  < 2e-16 ***
## a      2.72598    0.70218   3.882 0.000104 ***
## b      1.86724    0.69894   2.672 0.007568 ** 
## c     37.80243    3.28134  11.520  < 2e-16 ***
## d      1.78115    0.50574   3.522 0.000431 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6372 on 7255 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (72 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 6 rows containing missing values (`geom_point()`).

plotting 2

222 - Midwest Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1   5044     2430.6                                 
## 2   4824     1032.0 220 1398.6  29.715 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 25434.95
## 2     2 20512.77
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.28597    0.25555   1.119    0.263    
## alpha  0.75519    0.04617  16.357  < 2e-16 ***
## a      1.83111    0.29293   6.251 4.43e-10 ***
## b      1.47931    0.28426   5.204 2.03e-07 ***
## c     48.40459    3.83860  12.610  < 2e-16 ***
## d      1.75565    0.31800   5.521 3.55e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4625 on 4824 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (1015 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 7 rows containing missing values (`geom_point()`).

plotting 2

223 - Central Interior Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1   8872     4043.6                                 
## 2   8730     2553.3 142 1490.3  35.883 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 41571.02
## 2     2 37049.42
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.43007    0.13891  -3.096  0.00197 ** 
## alpha  0.61024    0.04351  14.025  < 2e-16 ***
## a      1.93241    0.63204   3.057  0.00224 ** 
## b      1.92173    0.62274   3.086  0.00204 ** 
## c     27.16683    1.99950  13.587  < 2e-16 ***
## d      1.71082    0.42581   4.018 5.92e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5408 on 8730 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (1274 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 6 rows containing missing values (`geom_point()`).

plotting 2

231 - Southeastern Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1  13446     7694.3                                 
## 2  13195     6476.7 251 1217.6  9.8832 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 70330.21
## 2     2 67127.47
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.16093    0.16727   6.941 4.09e-12 ***
## alpha  0.88288    0.02005  44.035  < 2e-16 ***
## a      2.23956    0.21991  10.184  < 2e-16 ***
## b      2.99104    0.21013  14.234  < 2e-16 ***
## c     17.69249    0.40947  43.209  < 2e-16 ***
## d      1.47519    0.10215  14.441  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7006 on 13195 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (316 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 30 rows containing missing values (`geom_point()`).

plotting 2

232 - Outer Coastal Plain Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1  13504     9637.4                                 
## 2  13221     8286.2 283 1351.2  7.6181 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 70511.84
## 2     2 67419.32
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.24859    0.19228   6.494 8.68e-11 ***
## alpha  0.87949    0.01947  45.179  < 2e-16 ***
## a      2.94637    0.11008  26.766  < 2e-16 ***
## b      2.14747    0.10445  20.561  < 2e-16 ***
## c     15.79158    0.43175  36.576  < 2e-16 ***
## d      0.89719    0.05082  17.654  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7917 on 13221 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (402 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 66 rows containing missing values (`geom_point()`).

plotting 2

234 - Lower Mississippi Riverine Forest

model selection 1

## Error in nls(fg2_1, data = G_234, start = c(tau = tau.start, a = a.start,  : 
##   Convergence failure: iteration limit reached without convergence (10)
##   model      AIC
## 1     1       NA
## 2     2 6970.096
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.49367    0.96311   1.551 0.121170    
## alpha  0.81742    0.08555   9.555  < 2e-16 ***
## a      3.25287    0.53212   6.113 1.29e-09 ***
## b      1.60897    0.49513   3.250 0.001185 ** 
## c     17.83130    2.52177   7.071 2.49e-12 ***
## d      0.68238    0.20357   3.352 0.000825 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8151 on 1316 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (66 observations deleted due to missingness)

summary

  • simple tau model: does not fit
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.90233, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -5.3394, p-value = 9.324e-08
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 5 rows containing missing values (`geom_point()`).

plotting 2

242 - Pacific Lowland Mixed Forest

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

251 - Prairie Parkland (Temperate)

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq  Df Sum Sq F value    Pr(>F)    
## 1   1888     981.71                                 
## 2   1773     366.60 115 615.11  25.868 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 9553.901
## 2     2 7360.029
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     0.7210     0.5093   1.416 0.157072    
## alpha   0.3998     0.1042   3.837 0.000129 ***
## a       0.0000     7.2404   0.000 1.000000    
## b       2.6602     7.2394   0.367 0.713317    
## c      29.1039     7.9249   3.672 0.000247 ***
## d       4.0010     6.4842   0.617 0.537291    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4547 on 1773 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (516 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.8207, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -8.0169, p-value = 1.085e-15
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 2 rows containing missing values (`geom_point()`).

plotting 2

255 - Prairie Parkland (Subtropical)

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value   Pr(>F)    
## 1    710    1009.74                               
## 2    667     883.04 43  126.7  2.2256 1.96e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 3560.631
## 2     2 3320.684
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     2.6192     2.2996   1.139 0.255118    
## alpha   0.6898     0.1649   4.183 3.26e-05 ***
## a       0.6449     0.5334   1.209 0.227051    
## b       1.9683     0.7959   2.473 0.013639 *  
## c      15.9536     1.9433   8.210 1.15e-15 ***
## d       1.3146     0.3969   3.312 0.000975 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.151 on 667 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (44 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.92514, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -3.8381, p-value = 0.000124
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 2 rows containing missing values (`geom_point()`).

plotting 2

261 - California Coastal Chaparral Forest and Shrub

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

262 - California Dry Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

263 - California Coastal Steppe - Mixed Forest and Redwood Forest

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

313 - Colorado Plateau Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

315 - Southwest Plateau and Plains Dry Steppe and Shrub

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

321 - Chihuahuan Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

322 - American Semidesert and Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

331 - Great Plains/Palouse Dry Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

332 - Great Plains Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

341 - Intermountain Semi-desert & Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

342 - Intermountain Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

411 - Everglades

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

M211 - Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   6765     1889.9                                
## 2   6741     1757.0 24 132.81   21.23 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 25839.54
## 2     2 25285.87
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.91021    0.21562   4.221 2.46e-05 ***
## alpha  0.63388    0.02892  21.920  < 2e-16 ***
## a      2.41117    0.18853  12.789  < 2e-16 ***
## b      0.72219    0.14719   4.907 9.48e-07 ***
## c     30.21358    2.16166  13.977  < 2e-16 ***
## d      1.07272    0.25230   4.252 2.15e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5105 on 6741 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (25 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

plotting 2

M221 - Eastern Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   8308     4863.8                                
## 2   8253     4464.8 55 398.96  13.408 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 40946.90
## 2     2 40030.35
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.37987    0.19436   1.955   0.0507 .  
## alpha  0.83193    0.05618  14.808  < 2e-16 ***
## a      2.83739    0.30722   9.236  < 2e-16 ***
## b      1.56703    0.25993   6.029 1.73e-09 ***
## c     26.79044    2.06992  12.943  < 2e-16 ***
## d      1.19017    0.24158   4.927 8.53e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7355 on 8253 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (56 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

predict and plot

## Warning: Removed 2 rows containing missing values (`geom_point()`).

plotting 2

M223 - Ozark Broadleaf Forest Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    890     537.81                                
## 2    883     515.61  7 22.204  5.4323 4.052e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 3747.851
## 2     2 3694.417
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     3.8819     1.9304   2.011 0.044634 *  
## alpha   0.8919     0.1493   5.976 3.32e-09 ***
## a       1.3980     0.3283   4.258 2.28e-05 ***
## b       0.8741     0.3169   2.758 0.005932 ** 
## c      32.1841     2.9862  10.778  < 2e-16 ***
## d       0.3997     0.1132   3.532 0.000433 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7641 on 883 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (7 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.9565, p-value = 1.42e-15
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -1.926, p-value = 0.05411
## alternative hypothesis: two.sided

predict and plot

plotting 2

M231 - Ouachita Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1000     560.60                                
## 2    987     494.92 13 65.684  10.076 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 4301.062
## 2     2 4134.044
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     4.7642     2.3590   2.020 0.043700 *  
## alpha   0.8568     0.1006   8.512  < 2e-16 ***
## a       1.2946     0.3543   3.654 0.000272 ***
## b       0.8329     0.2994   2.782 0.005497 ** 
## c       9.3076     4.1644   2.235 0.025636 *  
## d       1.2870     0.5699   2.258 0.024155 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7081 on 987 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (13 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.96666, p-value = 2.644e-14
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -6.1024, p-value = 1.045e-09
## alternative hypothesis: two.sided

predict and plot

plotting 2

M242 - Cascade Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   3140     2646.7                                
## 2   3127     2508.0 13 138.76  13.308 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 17325.39
## 2     2 17112.54
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -1.58025    0.29550  -5.348 9.54e-08 ***
## alpha  1.00476    0.07301  13.761  < 2e-16 ***
## a      6.39200    0.59569  10.730  < 2e-16 ***
## b      4.91557    0.91371   5.380 8.01e-08 ***
## c     34.09178    1.56232  21.821  < 2e-16 ***
## d      0.33514    0.05330   6.288 3.67e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8956 on 3127 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (91 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.92845, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -13.966, p-value < 2.2e-16
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 14 rows containing missing values (`geom_point()`).

plotting 2

M261 - Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1681     601.84                                
## 2   1668     585.67 13 16.172   3.543 1.672e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 8778.075
## 2     2 8686.805
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    -2.4098     0.2443  -9.864  < 2e-16 ***
## alpha   0.6694     0.1103   6.071 1.57e-09 ***
## a       7.0661     0.6834  10.340  < 2e-16 ***
## b       7.5903     1.4855   5.109 3.60e-07 ***
## c      31.6516     1.0663  29.684  < 2e-16 ***
## d       0.2004     0.0429   4.671 3.24e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5926 on 1668 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (303 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.90125, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -3.018, p-value = 0.002544
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 9 rows containing missing values (`geom_point()`).

plotting 2

M262 - Califormia Coastal Range = Coniferous Forest - Open woodland Shrub Meadow

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

M313 - Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    360     174.32                                
## 2    359     168.10  1 6.2251  13.295 0.0003055 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 1014.707
## 2     2 1003.435
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    -2.4926     0.3002  -8.304 2.08e-15 ***
## alpha   0.5805     0.1477   3.931 0.000101 ***
## a       0.0000     5.0587   0.000 1.000000    
## b       3.3501     5.1050   0.656 0.512094    
## c      61.8681    17.6811   3.499 0.000526 ***
## d       2.0796     2.2905   0.908 0.364526    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6843 on 359 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (2 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.94781, p-value = 4.677e-10
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -1.2182, p-value = 0.2232
## alternative hypothesis: two.sided

predict and plot

plotting 2

M331 - Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1736     1579.8                                
## 2   1719     1424.7 17 155.19  11.014 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 5243.073
## 2     2 5045.153
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.59259    0.66774  -0.887 0.374960    
## alpha  0.59000    0.06342   9.303  < 2e-16 ***
## a      0.40569    0.52551   0.772 0.440223    
## b      1.39910    0.54744   2.556 0.010683 *  
## c     50.60447    3.98505  12.699  < 2e-16 ***
## d      1.80430    0.53739   3.358 0.000804 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9104 on 1719 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (31 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.85792, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -5.2459, p-value = 1.556e-07
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 7 rows containing missing values (`geom_point()`).

plotting 2

M332 - Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   2527     1847.2                                
## 2   2485     1678.8 42 168.39  5.9346 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 9333.635
## 2     2 9043.297
## Warning in `[<-.data.frame`(`*tmp*`, nls.param.df2$Code == "M332", , value =
## structure(list(: provided 26 variables to replace 25 variables
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.64336    0.53013  -1.214 0.225020    
## alpha  0.83163    0.05528  15.044  < 2e-16 ***
## a      0.72335    0.62864   1.151 0.249979    
## b      1.82606    0.66347   2.752 0.005961 ** 
## c     61.15631    5.70900  10.712  < 2e-16 ***
## d      2.11186    0.57703   3.660 0.000258 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8219 on 2485 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (121 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.88303, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -6.6064, p-value = 3.938e-11
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 28 rows containing missing values (`geom_point()`).

plotting 2

M333 - Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1699     872.98                                
## 2   1670     777.35 29 95.626   7.084 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 6978.661
## 2     2 6735.140
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.09865    0.86145   0.115    0.909    
## alpha  0.85743    0.05817  14.741  < 2e-16 ***
## a      1.42786    0.29092   4.908 1.01e-06 ***
## b      2.32526    0.45714   5.087 4.06e-07 ***
## c     49.89803    2.07205  24.081  < 2e-16 ***
## d      1.08912    0.09906  10.995  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6823 on 1670 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (77 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.93058, p-value < 2.2e-16
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -5.5801, p-value = 2.403e-08
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 14 rows containing missing values (`geom_point()`).

plotting 2

M334 - Black Hills Coniferous Forest

model selection 1

## Error in nls(fg2_1, data = G_M334, start = c(tau = tau.start, a = a.start,  : 
##   Convergence failure: singular convergence (7)
##   model      AIC
## 1     1       NA
## 2     2 1382.769
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## tau     -3.4667     0.1424 -24.338  < 2e-16 ***
## alpha    0.7898     0.1840   4.293 2.28e-05 ***
## a        0.0000  1103.7169   0.000    1.000    
## b        4.0869  1103.4463   0.004    0.997    
## c       55.7353   113.2223   0.492    0.623    
## d        4.9145   686.6947   0.007    0.994    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4695 on 349 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (104 observations deleted due to missingness)

summary

  • simple tau model: does not fit
  • alpha model: fits

plot residuals

## 
## ------
##  Shapiro-Wilk normality test
## 
## data:  stdres
## W = 0.91466, p-value = 2.652e-13
## 
## 
## ------
## 
##  Runs Test
## 
## data:  as.factor(run)
## Standard Normal = -1.8005, p-value = 0.07178
## alternative hypothesis: two.sided

predict and plot

## Warning: Removed 1 rows containing missing values (`geom_point()`).

plotting 2

M341 - Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

plot residuals

## [1] "cannot plot residuals"

predict and plot

## [1] "cannot plot data with prediction"

plotting 2


Fitted parameters

Best / selected models by ecoprovince

Code Ecoregion Sel.Mod
211 Northeastern Mixed Forest 2
212 Laurentian Mixed Forest 2
221 Eastern Broadleaf Forest 2
222 Midwest Broadleaf Forest 2
223 Central Interior Broadleaf Forest 2
231 Southeastern Mixed Forest 2
232 Outer Coastal Plain Mixed Forest 2
234 Lower Mississippi Riverine Forest 2
242 Pacific Lowland Mixed Forest NA
251 Prairie Parkland (Temperate) 2
255 Prairie Parkland (Subtropical) 2
261 California Coastal Chaparral Forest and Shrub NA
262 California Dry Steppe NA
263 California Coastal Steppe - Mixed Forest and Redwood Forest NA
313 Colorado Plateau Semi-Desert NA
315 Southwest Plateau and Plains Dry Steppe and Shrub NA
321 Chihuahuan Semi-Desert NA
322 American Semidesert and Desert NA
331 Great Plains/Palouse Dry Steppe NA
332 Great Plains Steppe NA
341 Intermountain Semi-Desert and Desert NA
342 Intermountain Semi-Desert NA
411 Everglades NA
M211 Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow 2
M221 Central Appalachian Broadleaf Forest - Coniferous Forest - Meadow 2
M223 Ozark Broadleaf Forest Meadow 2
M231 Ouachita Mixed Forest 2
M242 Cascade Mixed Forest 2
M261 Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow 2
M262 California Coastal Range Coniferous Forest - Open Woodland - Shrub - Meadow NA
M313 Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow 2
M331 Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow 2
M332 Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow 2
M333 Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow 2
M334 Black Hills Coniferous Forest 2
M341 Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow NA

table by ecoprovince

Code Ecoregion region n.obs n.plots tau tau.variance tau.2.5 tau.97.5 alpha alpha.variance alpha.2.5 alpha.97.5 a a.2.5 a.97.5 b b.2.5 b.97.5 c c.2.5 c.97.5 d d.2.5 d.97.5
211 Northeastern Mixed Forest east 6877 2876 0.3259037 0.0322117 -0.0259256 0.6777330 0.6470960 0.0011462 0.5807292 0.7134629 0.0000000 -4.7842150 4.784215 3.4526191 -1.3104633 8.215701 31.418283 27.208885 35.62768 2.7742751 0.4658626 5.0826875
212 Laurentian Mixed Forest east 22715 9499 1.7565265 0.0395794 1.3665749 2.1464782 0.7606008 0.0004741 0.7179210 0.8032805 0.8604044 0.5826362 1.138173 1.4097375 1.1375575 1.681918 21.809974 20.738898 22.88105 1.9678736 1.6568644 2.2788829
221 Eastern Broadleaf Forest east 7333 3571 -0.5904555 0.0205932 -0.8717637 -0.3091473 0.7647928 0.0016125 0.6860759 0.8435098 2.7259833 1.3494989 4.102468 1.8672377 0.4971107 3.237365 37.802432 31.370059 44.23481 1.7811498 0.7897481 2.7725515
222 Midwest Broadleaf Forest east 5845 2589 0.2859672 0.0653067 -0.2150307 0.7869651 0.7551947 0.0021316 0.6646816 0.8457078 1.8311146 1.2568301 2.405399 1.4793082 0.9220371 2.036579 48.404588 40.879181 55.93000 1.7556482 1.1322300 2.3790665
223 Central Interior Broadleaf Forest east 10010 3864 -0.4300667 0.0192949 -0.7023555 -0.1577778 0.6102431 0.0018932 0.5249506 0.6955356 1.9324112 0.6934665 3.171356 1.9217261 0.7010148 3.142437 27.166828 23.247331 31.08633 1.7108213 0.8761331 2.5455096
231 Southeastern Mixed Forest east 13517 6193 1.1609328 0.0279784 0.8330644 1.4888011 0.8828828 0.0004020 0.8435823 0.9221832 2.2395602 1.8085049 2.670615 2.9910358 2.5791515 3.402920 17.692491 16.889878 18.49510 1.4751865 1.2749547 1.6754184
232 Outer Coastal Plain Mixed Forest east 13629 6626 1.2485936 0.0369726 0.8716922 1.6254949 0.8794887 0.0003790 0.8413311 0.9176462 2.9463741 2.7306069 3.162141 2.1474654 1.9427364 2.352194 15.791585 14.945296 16.63787 0.8971916 0.7975747 0.9968085
234 Lower Mississippi Riverine Forest east 1388 778 1.4936716 0.9275815 -0.3957277 3.3830708 0.8174228 0.0073191 0.6495898 0.9852558 3.2528661 2.2089608 4.296771 1.6089700 0.6376478 2.580292 17.831300 12.884170 22.77843 0.6823767 0.2830100 1.0817435
242 Pacific Lowland Mixed Forest pacific 83 83 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
251 Prairie Parkland (Temperate) east 2295 906 0.7210120 0.2594249 -0.2779536 1.7199776 0.3998419 0.0108569 0.1954814 0.6042023 0.0000000 -14.2005834 14.200583 2.6602066 -11.5383913 16.858805 29.103890 13.560728 44.64705 4.0009667 -8.7164867 16.7184201
255 Prairie Parkland (Subtropical) east 717 319 2.6192087 5.2881553 -1.8961158 7.1345333 0.6898464 0.0271917 0.3660624 1.0136304 0.6449176 -0.4023978 1.692233 1.9683297 0.4056346 3.531025 15.953585 12.137853 19.76932 1.3145518 0.5352792 2.0938244
261 California Coastal Chaparral Forest and Shrub pacific 25 25 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
262 California Dry Steppe pacific 0 0 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
263 California Coastal Steppe - Mixed Forest and Redwood Forest pacific 163 161 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
313 Colorado Plateau Semi-Desert interior west 218 218 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
315 Southwest Plateau and Plains Dry Steppe and Shrub interior west 4 4 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
321 Chihuahuan Semi-Desert interior west 9 9 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
322 American Semidesert and Desert interior west 3 3 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
331 Great Plains/Palouse Dry Steppe interior west 331 255 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
332 Great Plains Steppe interior west 232 128 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
341 Intermountain Semi-Desert and Desert interior west 66 64 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
342 Intermountain Semi-Desert interior west 124 123 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
411 Everglades east 96 63 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
M211 Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow east 6772 3006 0.9102083 0.0464909 0.4875300 1.3328866 0.6338783 0.0008363 0.5771893 0.6905674 2.4111710 2.0415978 2.780744 0.7221916 0.4336594 1.010724 30.213582 25.976035 34.45113 1.0727185 0.5781330 1.5673041
M221 Central Appalachian Broadleaf Forest - Coniferous Forest - Meadow east 8315 3810 0.3798738 0.0377742 -0.0011127 0.7608604 0.8319284 0.0031565 0.7217958 0.9420611 2.8373866 2.2351488 3.439624 1.5670342 1.0574968 2.076572 26.790440 22.732867 30.84801 1.1901696 0.7166178 1.6637213
M223 Ozark Broadleaf Forest Meadow east 896 349 3.8819227 3.7264172 0.0932287 7.6706166 0.8919471 NA 0.5990109 1.1848834 1.3979504 0.7536441 2.042257 0.8741302 0.2521252 1.496135 32.184123 26.323219 38.04503 0.3997004 0.1776253 0.6217755
M231 Ouachita Mixed Forest east 1006 495 4.7641593 5.5649730 0.1348893 9.3934294 0.8567699 0.0101303 0.6592587 1.0542812 1.2946131 0.5992694 1.989957 0.8329473 0.2455061 1.420388 9.307643 1.135637 17.47965 1.2869754 0.1685609 2.4053899
M242 Cascade Mixed Forest pacific 3224 3207 -1.5802457 0.0873175 -2.1596302 -1.0008612 1.0047559 0.0053311 0.8615956 1.1479162 6.3920015 5.2240166 7.559986 4.9155744 3.1240373 6.707111 34.091781 31.028497 37.15506 0.3351390 0.2306335 0.4396445
M261 Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow pacific 1977 1807 -2.4098383 0.0596808 -2.8889984 -1.9306782 0.6693505 0.0121571 0.4530890 0.8856120 7.0660557 5.7257203 8.406391 7.5902562 4.6765352 10.503977 31.651571 29.560209 33.74293 0.2003562 0.1162179 0.2844944
M262 California Coastal Range Coniferous Forest - Open Woodland - Shrub - Meadow interior west 30 26 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
M313 Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow interior west 367 367 -2.4926099 0.0901117 -3.0829541 -1.9022657 0.5805356 0.0218076 0.2901208 0.8709505 0.0000000 -9.9484618 9.948462 3.3501089 -6.6894500 13.389668 61.868141 27.096554 96.63973 2.0796397 -2.4249124 6.5841918
M331 Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow interior west 1756 1756 -0.5925858 0.4458749 -1.9022516 0.7170800 0.5899990 0.0040224 0.4656051 0.7143929 0.4056943 -0.6250198 1.436408 1.3990983 0.3253788 2.472818 50.604467 42.788414 58.42052 1.8042968 0.7502961 2.8582975
M332 Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow interior west 2612 2602 -0.6433640 0.2810415 -1.6829128 0.3961848 0.8316342 0.0030560 0.7232333 0.9400351 0.7233522 -0.5093529 1.956057 1.8260584 0.5250439 3.127073 61.156314 49.961419 72.35121 2.1118589 0.9803577 3.2433601
M333 Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow interior west 1753 1742 0.0986515 0.7421010 -1.5909896 1.7882926 0.8574276 0.0033834 0.7433405 0.9715147 1.4278560 0.8572552 1.998457 2.3252552 1.4286291 3.221881 49.898029 45.833939 53.96212 1.0891217 0.8948317 1.2834117
M334 Black Hills Coniferous Forest interior west 459 181 -3.4667353 0.0202887 -3.7468811 -3.1865895 0.7897504 0.0338387 0.4279547 1.1515460 0.0000000 -2170.7734030 2170.773403 4.0869419 -2166.1541718 2174.328056 55.735252 -166.948614 278.41912 4.9144979 -1345.6661082 1355.4951039
M341 Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow interior west 220 220 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA

parameter variance co-variance

## png 
##   2

plot tau

map

## OGR data source with driver: ESRI Shapefile 
## Source: "C:\Users\hogan.jaaron\Dropbox\FIA_R\Mapping\S_USA.EcoMapProvinces\S_USA.EcoMapProvinces.shp", layer: "S_USA.EcoMapProvinces"
## with 37 features
## It has 17 fields
## Integer64 fields read as strings:  PROVINCE_ PROVINCE_I
## Warning: package 'ggnewscale' was built under R version 4.2.1
## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation ideoms with `aes()`
## Warning: The `size` argument of `element_line()` is deprecated as of ggplot2 3.4.0.
## ℹ Please use the `linewidth` argument instead.
## Warning in grid.Call(C_stringMetric, as.graphicsAnnot(x$label)): font family not
## found in Windows font database

## Warning in grid.Call(C_stringMetric, as.graphicsAnnot(x$label)): font family not
## found in Windows font database
## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

plot alpha (biomass growth compensation effect)

plot a coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).

plot b coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).

plot c coefficient

## Warning: Removed 1 rows containing missing values (`geom_hline()`).
## Warning: Removed 16 rows containing missing values (`geom_point()`).

plot d coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).

Caclulations - weighted averages

tau (productivity trend (in %) 2000-2022)

##          region weighted.tau weighted.tau.std_Error 95 % CI, upper
## 1     entire US   0.46776517             0.07376303    0.612340704
## 2       pacific  -0.16266596             0.01804796   -0.127291955
## 3          east   0.70095388             0.05885088    0.816301604
## 4 interior west  -0.07052274             0.04064270    0.009136941
##   95 % CI, lower
## 1      0.3231896
## 2     -0.1980400
## 3      0.5856061
## 4     -0.1501824

alpha (biomass growth compensation effect)

##          region weighted.alpha weighted.alpha.std_Error 95 % CI, upper
## 1     entire US     0.75849014              0.010377384     0.77882982
## 2       pacific     0.07650874              0.005307716     0.08691186
## 3          east     0.59480677              0.008086378     0.61065607
## 4 interior west     0.08717463              0.003758821     0.09454192
##   95 % CI, lower
## 1     0.73815047
## 2     0.06610561
## 3     0.57895747
## 4     0.07980735